This could be caused by a few different things. Make sure the device has enough charge to power on. If the device is charged and the issue still persists, it could be a faulty power switch. This can be fixed by replacing the power switch.
Another cause could be the battery. The capacity of the battery is reduced after each charge cycle. This is a fairly old device and it is possible that the battery is no longer functional. You may need to replace the battery.
Finally, this issue could just be a result of the display not turning on. If this is the case, you will need to replace the device's display.
Pandigital Novel PRD07T10WWH7 Troubleshooting